A note on the points with dense orbit under the expansions of different bases

It was conjectured by Furstenberg that for any x∈[0,1]∖Q,dimH{2nx(mod1):n≥1}¯+dimH{3nx(mod1):n≥1}¯≥1,where dimH denotes the Hausdorff dimension and A¯ denotes the closure of a set A. When x is a normal number, the above result holds trivially. In this note, we are aiming at giving explicit non-normal numbers for which the above dimensional formula holds.